Existence globale et diffusion pour l’équation de Schrödinger non linéaire répulsive cubique sur $mathbb{R}^3$ en dessous l’espace d’énergie

Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T.

doi:10.5802/jedp.608

Existence globale et diffusion pour l’équation de Schrödinger non linéaire répulsive cubique sur

m a t h b b R^{3}

en dessous l’espace d’énergie

Colliander, J. ; Keel, M. ; Staffilani, G. ; Takaoka, H. ; Tao, T.

Journées équations aux dérivées partielles (2002), article no. 10, 15 p.

Résumé
Abstract

Nous profilons une démonstration de l’existence globale et diffusion pour l’équation de Schrödinger nonlinéaire répulsive cubique avec données à $H^{s} (ℝ^{3})$ pour $s > \frac{4}{5}$ . Le raisonnement utilise une estimation nouvelle de type de Morawetz. Nous détaillerons la démonstration ailleurs.

We sketch a proof of global existence and scattering for the defocusing cubic nonlinear Schrödinger equation in $H^{s} (ℝ^{3})$ for $s > \frac{4}{5}$ . The proof uses a new estimate of Morawetz type.

DOI : 10.5802/jedp.608

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     author = {Colliander, J. and Keel, M. and Staffilani, G. and Takaoka, H. and Tao, T.},
     title = {Existence globale et diffusion pour l{\textquoteright}\'equation de {Schr\"odinger} non lin\'eaire r\'epulsive cubique sur $mathbb{R}^3$ en dessous l{\textquoteright}espace d{\textquoteright}\'energie},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {10},
     pages = {1--15},
     publisher = {Universit\'e de Nantes},
     year = {2002},
     doi = {10.5802/jedp.608},
     mrnumber = {1968206},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.608/}
}

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Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Existence globale et diffusion pour l’équation de Schrödinger non linéaire répulsive cubique sur $mathbb{R}^3$ en dessous l’espace d’énergie. Journées équations aux dérivées partielles (2002), article  no. 10, 15 p. doi : 10.5802/jedp.608. http://www.numdam.org/articles/10.5802/jedp.608/

Bibliographie
Cité par

[1] M. Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations Ann. Math. 118, (1983), 187-214. | MR | Zbl

[2] J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity Intern. Mat. Res. Notices, 5, (1998), 253-283. | MR | Zbl

[3] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12, (1999), 145-171. | MR | Zbl

[4] J. Bourgain. Scattering in the energy space and below for 3D NLS, Jour. D'Anal. Math., 75:267-297, 1998. | MR | Zbl

[5] J. Bourgain, Global solutions of nonlinear Schrödinger equations American Math. Society, Providence, R.I., 1999. | MR | Zbl

[6] J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, GAFA, 3 (1993), 107-156. | MR | Zbl

[7] T. Cazenave, F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in H 1. Manuscripta Math. 61 (1988), 477-494. | MR | Zbl

[8] J. Colliander, G. Staffilani, H. Takaoka, Global wellposedness for KdV below L 2, Math. Res. Lett. 6 (1999), no. 5-6, 755-778. | MR | Zbl

[9] J. Colliander, J. Delort, C. Kenig, G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), no. 8, 3307-3325. | MR | Zbl

[10] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Global well-posedness for KdV in Sobolev spaces of negative index, Electronic Jour. Diff. Eq. 2001 (2001), No 26, 1-7. | MR | Zbl

[11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Sharp Global WellPosedness of KdV and Modified KdV on the R and T, submitted to Jour. Amer. Math. Soc. | MR | Zbl

[12] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Multilinear estimates for periodic KdV equations, and applications, preprint. | MR

[13] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Global well-posedness for the Schrödinger equations with derivative, Siam Jour. Math. Anal., 33 (2001), 649-669. | MR | Zbl

[14] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T.Tao, A refined global wellposedness result for Schrödinger equations with derivative, to appear in Siam Jour. of Math. Anal.. | MR | Zbl

[15] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Almost conservation laws and global rough solutions to a Nonlinear Schrödinger Equation, to appear in Math. Res. Letters. X-13 | MR | Zbl

[16] G. Fonseca, F. Linares, and G. Ponce. Global well-posedness of the modified Korteweg-de Vries equation, Comm. Partial Differential Equations, 24 (1999), 683-705. | MR | Zbl

[17] J. Ginibre and G. Velo. Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl., 9, (1985), 363-401. | MR | Zbl

[18] J. Ginibre, G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 123 (1989), 535-573. | MR | Zbl

[19] M. Keel, and T. Tao, Global well-posedness for large data for the MaxwellKlein-Gordon equation below the energy norm, preprint.

[20] M. Keel, T. Tao, Local and global well-posedness of wave maps on R 1+1 for rough data, Intl. Math. Res. Notices 21 (1998), 1117-1156. | MR | Zbl

[21] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980. | MR | Zbl

[22] C. Kenig, G. Ponce, L. Vega; Global well-posedness for semi-linear wave equations. Comm. Partial Diff. Eq. 25 (2000), 1741-1752. | MR | Zbl

[23] J. Lin, W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, Journ. Funct. Anal. 30, (1978), 245-263. | MR | Zbl

[24] C. Morawetz, Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. A 306 (1968), 291-29 | MR | Zbl

[25] C. Morawetz, W. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972), 1-31. | MR | Zbl

[26] I.E. Segal, Space-time decay for solutions of wave equations, Adv. Math. 22 (1976), 304-311. | MR | Zbl

[27] R.S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705-774. | Zbl

[28] H. Takaoka, Global well-posedness for the Schrödinger equations with derivative in a nonlinear term and data in low order Sobolev space, Electronic Jour. Diff. Eq., 42 (2001), 1-23. | MR | Zbl

[29] H. Takaoka, N. Tzvetkov, Global low regularity solutions for KadomtsevPetviashvili equation, Internat. Math. Res. Notices, 2001, No. 2, 77-114. | Zbl

[30] T. Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), 443-544. | MR | Zbl

[31] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987) | MR | Zbl

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